# Why does Sage say my Groebner basis is 1?

Consider the following program

A.<a1, a2, a3, l2, l3> = PolynomialRing(QQ)
F = A.fraction_field()
F.inject_variables()
R.<c2, s2, c1, s1> = PolynomialRing(F, order = 'lex')
I = R.ideal(l3*s1*s2 + l2*c1 - a1, l3*s1*c2 - l2*s1 - a2, l3*c2 - a3, c1^2 + s1^2 - 1, c2^2 + s2^2 - 1)
I.groebner_basis()


So $a_1, a_2, a_3, l_2, l_3$ are parameters, and I am trying to solve the system of equations

$$a_1 = l_3s_1s_2 + l_2c_1$$

$$a_2 = l_3s_1c_2 - l_2s_1$$

$$a_3 = l_3c_2$$

$$c_1^2 + s_1^1 = 1$$

$$c_2^2 + s_2^2 = 1$$

Now this gives as result 1, meaning that there is no solution to this system. However, for $a_1 = l_2$, $a_2 = 0$ and $a_3 = l_3$ we do have a solution, namely $c_1 = 1, s_1 = 0, c_2 = 1, s_2 = 0$.

So is something in the programming going wrong, or am I misunderstanding something on the algebraic geometry part?

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If you work over the fraction field of the parameters, you implicitly assume that all the parameters are nonzero, and even more: that all nontrivial polynomial expressions in the parameters are nonzero (because they all may appear as denominators). Concretely: R.one().lift(I) gives a list of coefficients of generators of I such that the linear combination equals 1. (In particular, it shows that the Groebner basis is correct.) Of course these coefficients are (formal) fractions. But the non-vanishing of the denominator of these fractions is in fact inconsistent with the original system of equations. Put another way: the parameters can not all be chosen arbitrarily; they must satisfy some consistency condition that follows from the system of equations (see below).

Better just work in a big polynomial ring:

sage: R.<c2, s2, c1, s1, a1, a2, a3, l2, l3> = PolynomialRing(QQ, order='lex')
sage: I = R.ideal(l3*s1*s2 + l2*c1 - a1, l3*s1*c2 - l2*s1 - a2, l3*c2 - a3, c1^2 + s1^2 - 1, c2^2 + s2^2 - 1)
sage: list(I.groebner_basis())


You get a big "triangular" system of equations that you can try to solve from the bottom up.

The last equation involves only the parameters, so it is a consistency condition (if it is not satisfied, then there is no solution). The second-to-last equation is $s_1a_3 - s_1l_2 - a_2=0$. If $a_3\neq l_2$ then you can solve it for $s_1$. If $a_3=l_2$ then $a_2=0$ and the remaining equation for $s_1$ is (at most) a quartic depending on the parameters, and so on.

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Thank you!

( 2020-06-05 06:59:33 -0500 )edit